Hyperbolic lattices: From Hofstadter Butterfly to Experimentally Realizable Cayley crystal decomposition

Hyperbolic lattices, characterized by negative curvature and non-commutative translations,
offer a rich playground for exploring exotic electronic states. This talk explores these systems
through a multifaceted approach, bridging theory and experiment. We begin with a concise
introduction to hyperbolic lattices and then move to the results to present curvature-dependent
Hofstadter butterfly spectrum in the presence of a magnetic field; we acknowledge the experimental
challenges in directly realizing these structures. We introduce an indirect approach that decomposes
the problem with hyperbolic lattices into two parts: curvature and non-commutative geometry. This
method breaks down hyperbolic lattices into curved Euclidean lattices (amenable to strain
engineering in graphene) and simpler non-Abelian Z 2 lattices generated by non-commuting
translations (Cayley crystals). In the first case, we investigate topological states in curved graphene,
leveraging Kitaev's real-space index to characterize their behavior. Finally, in the end, I will present
our very recent findings (still under progress), revealing two distinct classes of states within these Z 2
lattices: Abelian states exhibiting conventional behavior and non-Abelian states experiencing a
surprising Hall drift motion under an electric field. This intriguing result suggests the presence of an
effective internal magnetic field in the non-Abelian sector, opening exciting avenues for investigating
novel physical phenomena.